Solve 9x^2-30x+43>0

$$9 { x }^{ 2 } -30x+43 > 0$$

Solve for x

$$x\in \mathrm{R}$$

Steps Using the Quadratic Formula

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$9x^{2}-30x+43=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 9 for a, -30 for b, and 43 for c in the quadratic formula.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 9\times 43}}{2\times 9}$$

Do the calculations.

$$x=\frac{30±\sqrt{-648}}{18}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 9x^{2}-30x+43 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$9\times 0^{2}-30\times 0+43=43$$

The value of the expression 9x^{2}-30x+43 is always positive. Inequality holds for x\in \mathrm{R}.

$$x\in \mathrm{R}$$